On the Sparsity Order of a Graph and Its Deficiency in Chordality

Abstract

Given a graph G on n nodes, let denote the cone consisting of the positive semidefinite matrices (with real or complex entries) having a zero entry at every off-diagonal position corresponding to a non edge of G. Then, the sparsity order of G is defined as the maximum rank of a matrix lying on an extreme ray of the cone . It is known that the graphs with sparsity order 1 are the chordal graphs and a characterization of the graphs with sparsity order 2 is conjectured in [1] in the real case. We show in this paper the validity of this conjecture. Moreover, we characterize the graphs with sparsity order 2 in the complex case and we give a decomposition result for the graphs with sparsity order in both real and complex cases. As an application, these graphs can be recognized in polynomial time. We also indicate how an inequality from [17] relating the sparsity order of a graph and its minimum fill-in can be derived from a result concerning the dimension of the faces of the cone .